Superluminal Motions of AGNs and GRBs at Multiple-Frequencies
aa r X i v : . [ a s t r o - ph . H E ] J a n Astronomy&Astrophysicsmanuscript no. ms˙SM˙v3 c (cid:13)
ESO 2018August 29, 2018
Superluminal Motions of AGNs and GRBs at Multiple-Frequencies
Zhi-Bin Zhang , Department of Physics, College of Physics, Guizhou University, Guiyang, Guizhou 550025, China.e-mail: [email protected] College of Physics and Engineering, Qufu Normal University, Qufu 273165, China.Preprint online version: August 29, 2018
ABSTRACT
Context.
Superluminal motion has been found to occur in many kinds of celestial bodies with the relativistic ejecta, especially forsome active galactic nuclei (AGNs), gamma-ray bursts (GRBs) and micro-quasars, although these objects are largely di ff erent in theirlifetimes, sizes and features. Aims.
To investigate the discrepancy in above inhomogeneous objects, we have compared the apparent superluminal motion of GRBswith AGNs, both of them being cosmological outbursts. In addition, properties of the superluminal motion between di ff erent sub-classes of AGNs (e.g., Radio galaxies, BL Lac objects, and Quasars) are also compared in detail.Particularly, we focus on two lowluminosity GRBs, namely 060218 and 170817A, that are ultra-long and short bursts associated with supernova and gravitational waverespectively. Methods.
For these, statistical methods including linear regression analysis, linear correlation and K-S test have been used. Antithesesand theoretical analysis of models are also adopted to contrast between AGNs and GRBs, as well as their individual subgroups ataspect of theory and observation.
Results.
The apparent transverse velocity ( β app ) of Swift and pre-Swift GRBs are tightly correlated with the Doppler factor δ as β app ∝ δ . , while all AGNs distribute around the line of β app = δ behaving a weak correlation of β app with δ . In contrast, β app ispositively correlated with Lorenz factor ( γ ) or γ (1 + z ) − , not for GRBs but for AGNs. β app and 1 + z are independent for neitherGRBs nor AGNs, but apparently exhibit a positive correlation of β app with 1 + z from AGNs to GRBs, showing a cosmological e ff ectof evolution with redshift. I also found that GRB 170817A and GRB 060218 are outliers of the above correlations. Conclusions.
Superluminal properties of GRBs are significantly di ff erent from those of AGNs. However, there are no obvious di ff er-ences between radio galaxies, BL Lac objects, and quasars in terms of their superluminal motion. In despite of radiation mechanism,beaming e ff ect and cosmological expanding would play the same roles on the superluminal motion for AGNs and GRBs. Key words. gamma-ray bursts–active galactic nuclei–apparent superluminal motion–cosmology
1. Introduction
The apparent superluminal phenomenon was first predicted by Rees (1966) that the transverse velocity of an object moving (ultra-)relativistically in some special directions may appear to exceed the speed of light. At first glance such a motion is quite counter-intuitive, whereas it does not violate the special relativity. It is essentially a geometric e ff ect or a light travel-time e ff ect in the frameof standard model (Chodorowski 2005).This motion is initially confirmed and discovered by using new technique of spectroscopy called as very long baseline in-terferometry (VLBI) for radio galaxies and quasars (Whitney et al. 1971, Cohen et al. 1971). So far, it has been known that thesuperluminal motion is not unique to quasars and radio galaxies, but also to other sources including micro-quasars and BL Lacobjects (e.g. Mirabel & Rodr´ıguez 1994, Fan, Xie & Wen 1996, Jorstad et al. 2001, Kellermann et al. 2004) and so on. This may bebecause jets or jet-like outflows are common among various kinds of astrophysical phenomena with di ff erent scales, such as sun,proto-stellar systems, isolated neutron stars, neutron star and black hole binaries, supermassive black holes (or AGNs) and GRBs(Zhang 2007). Once the apparent velocity is measurable, one can have opportunity to learn the geometry and the underlying physicson the formation, ejection and acceleration of jets (Ghisellini & Celotti A 2002).AGNs are generally thought to originate from supermassive black holes located in their centers. Classification is a very importantstrategy to comprehend their basic physics. They can be classified into several groups or subgroups by means of some diagnosticdiagrams, e.g. Seyfert I and II galaxies, Quasars, Blazars (BL lac Objects and OVVs), radio galaxies, and LINER. On the whole,Blazars, radio galaxies and part of Quasars are radio-loud, the remainder is radio-quiet. The radio-loud AGNs typically exhibitstrong radio radiation, variable, jetted outflows, and X-ray emission and UV excess. This is surprisingly similar to what have beenobserved in GRBs. Moreover, these radio-loud sources together with GRBs have a common property that the ejecta from their coresmove very close to the speed of light outwards and both of them are similarly cosmological outbursts. Till now, only two GRBs, e.g.030329 and 170817A, are promised to have the detectably superluminal phenomenon in radio bands (Dado, Dar & De Rujula 2004,2018; Taylor et al. 2004). Although the superluminal motions of GRBs in high energy afterglows even prompt γ -rays are invisiblecurrently, the superluminal e ff ect for some GRBs would do exist in theory, as long as the synergetic conditions such as viewingangles and boosting lorentz factors of outflows were satisfied (see Fig. 1). This motivates us to give a comparative study between Zhi-Bin Zhang: Superluminal Motions of AGNs and GRBs at Multiple-Frequencies
GRBs and radio-loud AGNs on the superluminal motion. Besides, GRBs in the pre-Swift and Swift eras have also been comparedto check the dependency of the apparent transverse velocity on di ff erent instruments. =2 =10 =5 =20 app (radian) Fig. 1.
The apparent transverse velocity β app vs. the viewing angle θ . Note that every β app > θ max = arccos β .
2. Superluminal motion
According to the relative geometry of ejecta to the observer, the apparent transverse velocity β app is generally defined by β app = β sin θ − β cos θ (1)in an unit of speed of light c , of which β represents the real speed of ejecta and θ is the viewing angle between the line of sight andthe bulk velocity outwards. Note that the β is also in the unit of c and associated with Lorentz factor γ by β = (1 − γ − ) . . For agiven γ , one can make a plot of β app versus γ , where shows a single peak profile at θ ≈ /γ (Cohen, Lister & Vermeulen 2004,Kellermann et al. 2007).In addition to β app , Doppler factor δ is also connected with θ and γ as δ = γ (1 − β cos θ ) , (2)In Eqs. (1) and (2), four independent variables, i.e. β app , β , θ and δ ), are involved. Provided two of them have been known inadvance, the rest two should be deduced immediately. For the radio observations to AGNs with measured redshift ( z ), β app is ingeneral measurable and determined by β app = µ D A c , (3)where µ is the observed proper motion in an unit of mas / yr, D A denotes the angular diameter distance of radio source. In practice,the physical luminosity distance, D L in the flux expression of f = L / (4 π D L ), is usually used for cosmological studies. D A and D L are tied with each other as D L = (1 + z ) D A , (4)Hence, β app can then be newly expressed as follows β app = µ D L c (1 + z ) , (5)The Doppler factor δ (Ghisellini et al. 1993) is estimated by δ = f ( α ) S m [ ln ( ν b /ν m ) S x φ + α d ν α x ν + α m ] / (4 + α ) · (1 + z ) ; (6) hi-Bin Zhang: Superluminal Motions of AGNs and GRBs at Multiple-Frequencies 3 where f ( α ) ≃ . α + .
14 is an experiential function decided by an index α of the enhancement ( ∝ δ α ) of the observed flux due tobeaming e ff ect, S m is the maximum flux at the frequency ν m in GHz. S x like S m in the unit of Jansky (Jy) represents the flux withinthe range of any other energy band ν x , other than the radio. ν b and φ d are the synchrotron high energy cuto ff and the size of core,respectively. As for AGNs, if only the redshift can be gotten from the radio spectra one always calculate β app and δ , and then β and θ . In principle, the superluminal phenomenon can be seen as long as the viewing angle θ is suitably small enough in comparisonwith the opening angle ∼ /γ of outflows due to the beaming action. For a given source, its opening angle is generally knowable ina direct or indirect way, which makes Eq. (1) degenerate into three scenarios in the following three cases of θ as:1. In the case of θ ≪ γ − , the observer is enveloped within the cone of jets. Some simplified forms of sin θ ∼ θ and cos θ ∼ β app ≃ γ θ (7)2. In the case of θ ∼ γ − , one can just see the edge of outflows under assumption that jets do not behave a significant expandingsideways. Relations of cos θ ∼ − γ − /
2, sin θ ∼ γ − and β = − γ − / β app will be reduced and reachits maximum as β app ≃ γ ≃ δ ≃ csc θ (8)3. In the case of θ ≫ γ − , the observer is located out of the cone of jets, which means sin θ ∼ θ , cos θ ∼ − θ / β ∼
1, thus β app ≃ /θ (9)
3. Samples
In view of above considerations, 8 radio galaxies, 15 BL Lac objects, and 46 quasars with superluminal observations by VLBI havebeen chosen from literatures (Hong et al. 1995, Jorstad et al. 2001, Kellermann et al. 2004), to constitute three independent AGNsamples table 1.For GRBs, present observations of radio afterglows hiding superluminal motion are very rare. Till now, ONLY GRB 030329and GRB 980425 are suggested to be the potential candidates (e.g. Dar & De Rujula 2000; Dado, Dar & De Rujula 2004). Thereason is that the angular resolution of current antennae is still too lower to measure the proper motion of GRB afterglows, exceptfor GRB 030329 ( Taylor et al. 2004). It will be more di ffi cult to describe the superluminal motion in the phase of prompt emissionsdue to lack of much higher resolution. This causes Eq.(3) to be impossible for deriving β app . Alternatively, Eq. (1) can be used toinvestigate the superluminal motion during the prompt γ -ray emissions if only the viewing angle θ can be conveniently measuredprovided the Lorentz factors of GRBs are roughly estimated by γ ≈ × E / iso , n − / ( T / s ) − / , (10)as suggested by Rees & M´esz´aros (1992). Here, it has been assumed that the jet opening angle θ j is equivalent to the viewingangle θ since the jetted outflow from us is much farther than that from its central engine, which will give the upper limits toestimate the apparent velocities of GRBs. As seen in Fig. 1, larger lorentz factors together with wider viewing angles will resultin an enhancement of the superluminal motion. Based on the consideration, the apparent superluminal motion of GRBs in theirprompt phase had been studied. For the sake of comparison, 27 Swift and 37 pre-Swift long GRBs with known redshift, durationtime and jet break time are taken from Zhang, Zhao & Zhang (2011). In addition, I pay particular attention to GRB 170817Awhose redshift, lorentz factor and viewing angle are estimated as z = . γ ≃
13 (Zou et al. 2018) and30 ◦ ≥ θ ≥ ◦ (Haggard et al. 2017; Ioka & Nakamura 2017), respectively. Using these measured values, one can easily obtain theapparent transverse velocity of GRB 170817A to be β app = . + . − . and the Doppler factor as δ = . + . − . .
4. Results
In this section, the main results related to superluminal motion are displayed for di ff erent kinds of cosmological sources. I payspecial attention to the comparison between AGNs and GRBs in both radio and γ energy bands. β app with δ , z and γ Fig. 2 shows a largely di ff erent population between GRBs and AGNs in the plane of β app versus δ . For pre-Swift and Swift longGRBs, however, they exhibit a similar positively correlated tendency. The dotted and dashed lines correspond to their best fits to theGRB data, following a very close power-law of β app ∼ δ . . Unlike GRBs, most AGNs locate within the range of θ ≥ . γ − androughly distribute along the line of β app ∝ δ with larger dispersion. A linear correlative analysis gives the correlation coe ffi cient of0.6 for all AGNs with a tiny chance probability of P < . β app obviously evolve with redshift for any kinds of these cosmologicalsources. Furthermore, Fig.3 shows the positive correlation of β app with redshift becomes more tight when all AGNs and GRBs arecombined to be a whole sample. The positive correlations demonstrate farther the sources observed larger the apparent transversevelocity. Based on the three relations between θ and 1 /γ in section 2, we see that β app ∼ γ and β app ∼ /θ when θ ∼ /γ and θ > /γ Zhi-Bin Zhang: Superluminal Motions of AGNs and GRBs at Multiple-Frequencies -1 -1 -1 -1 = -1 =0.1 -1 app GRB170817AGRB060218
Fig. 2.
The apparent transverse velocity β app vs. the Doppler factor δ for AGNs and GRBs in our samples. Circles:
Swift GRBs; squares: pre-Swift GRBs; stars: quasars; triangles:
BL Lac objects; diamonds:
Radio galaxies. Thick, thin and dash-dotted linesdenote the relations of β app = δ , β app = . δ and β app = constant , respectively. The dotted and dashed lines correspond to their bestfits to the GRB data. All the symbols defined here are same to those in the Figs. 3, 4 and 5. -1 a pp G R B A G R B Fig. 3.
The apparent transverse velocity β app vs. cosmological inflation factor of 1 + z in the logarithmic scale.for AGNs and GRBs, respectively. γ and β app are tightly correlated for the di ff erent radio-loud AGNs, while there is no any relationfor LGRBs. This is because the viewing angle θ is far large than 1 /γ , which cause β app becomes only dependent on θ . However,AGNs with θ ∼ /γ seen from Fig.2 make them locate around the line of β app ∼ γ .Plot of β app vs. γ in Fig.4a shows how the apparent transverse velocity is a ff ected by Lorentz factor for all the radio-loud AGNsand LGRBs. It is interestingly found that the β app is positively correlated with γ , while this trend disappears for GRBs. Fig.4bindicates that the intrinsic observations, e.g. some typically comoving timescale in source frame, reduced for cosmological inflationand Doppler e ff ect still hold above similar properties for di ff erent AGNs and GRBs. This means AGNs and GRBs are basicallydistinct cosmological sources in nature.In addition, one can find in Figs. 2, 3 and 4 that radio galaxies are distributed in the small end of β app , largely di ff erent fromBL Lac objects and Quasars, much less the LGRBs. This means the radio galaxies with lower radio luminosity are very special andmay thus result from a distinct origin, compared with other radio-loud AGNs such as BL Lac objects and Quasars. hi-Bin Zhang: Superluminal Motions of AGNs and GRBs at Multiple-Frequencies 5 -1 GRB060218GRB170817A app (a)
GRB170817A GRB060218 10 -1 ( + z ) - app (b) Fig. 4.
Correlations of the apparent velocity β app with the Lorentz factor γ in panel (a) and the modificatory factor γ (1 + z ) − forthe e ff ects of Doppler boosting and cosmological expansion in panel (b), respectively. (radian) GRB060218GRB170817A
Fig. 5.
The Lorentz factor γ vs. the viewing angle θ . Solid line corresponds to δ = dashed line represents γ = / sin θ ; and dottedline describes β app = γ and θ In Fig.5, we compare these sources in the γ vs. θ plane. The top-right region of δ = δ < δ >
1. Similarly, the bottom-left of β app = β app < β app = β app >
1. Very clearly, most of AGNs and GRBs reside between the solid line and the dotted one, in the area of β app > δ >
1. The result on AGNs is extremely consistent with that suggested by Ghisellini (1993) in theory. But we again find the radiogalaxies with lower lorentz factor compared to other AGNs locate at the larger end of θ . Another interesting phenomenon is GRBswith smaller viewing angle distribute at the larger end of γ . This indicates that the beaming e ff ect could be really not very importantfor the radio galaxies, unlike for GRBs, BL Lac objects and Quasars. A majority of AGNs follows γ = / sin θ , that is β app = βδ ,while GRBs cluster along the line of δ =
1. In fact, the locations of GRBs might move towards the short end of θ slightly if the jetevolution is considered. Zhi-Bin Zhang: Superluminal Motions of AGNs and GRBs at Multiple-Frequencies
It is well-known that GRB 060218 is a typical low luminosity burst ( T ∼ s ) connected with SN 2006aj, a supernovae Ib / c (e.g.Mirabal et al. 2006; Li 2007). Recently, Zhang et al. (2018a) noticed that GRB 060218, together with 980425 and 031203, deviatedfrom the normal evolution curves of long bursts in the plane of peak flux density versus redshift at di ff erent radio frequencies. GRB170817A is a peculiarly newly-discovered short GRB ( T ∼ s ) associated with gravitational wave (GW), which was reported asthe first electromagnetic counterpart of a GW event originated from binary Neutron stars (Abbott et al. 2016a,b). With more andmore short bursts accumulated with redshift and peak energy, their spectrum-energy relations can be built as long bursts did before.Excitingly, three spectrum-energy correlations have been proposed by Zhang et al. (2018b) and the GRB 170817A, unlike othershort and long bursts, is identified as an outlier to all the three energy relations of short GRBs. It is interesting to note here that GRB170817A is also found to be very di ff erent from long bursts but similar to GRB 060218 regarding the superluminal motion. Thismay hint that those low luminosity GRBs could hold the consistent observations related with the superluminal motions.
5. Conclusions and discussions
Our main results are summarized and concluded in the following:1. The apparent transverse velocity β app is found to be correlated with δ as β app ∝ δ for LGRBs and β app ∝ δ . for radio-loudAGNs.2. Radio galaxies are special AGNs with lower redshift, luminosity, γ and δ , which makes it a very di ff erent radio-loud sources.3. Figs. 2,4 and 5 seem to demonstrate GRBs and AGNs may have di ff erent radiation mechanisms and Doppler (or beaming)e ff ect.However, their cosmological evolution might be same as shown in Fig. 3 in respect of the superluminal motion.4. As low luminosity GRBs, 060218 and 170817A are obvious violators to the above relation of both long GRBs and three typesof AGNs and have very similar behaviors although they are largely di ff erent in duration time T .As suggested by Kellermann et al. (2004), radio jets that are also strong gamma-ray sources detected by EGRET appear to havesignificantly faster speeds than non-EGRET sources, which is much consistent with the idea that gamma-ray sources have largerDoppler factors than nonCgamma-ray sources. Kellermann et al. (2004) also pointed that there is a systematic decrease in apparentvelocity with increasing wavelength, probably because the observations at di ff erent wavelengths sample built from di ff erent partsof the jet structure. It is required to test whether this is also true for GRBs because the current sample with with superluminalmotion for GRB radio afterglows are very limited, ONLY GRB 030329 and possibly GRB 980425. In such a case, observationalinvestigations on superluminal motion of GRBs are particularly expected in the near future.The typically extragalactic sources with high redshift are usually useful for studying the formation and evolution of the earlyuniverse. So far, the farthest galaxy is found to have a spectroscopic redshift of z = + z = .
28, which was taken among follow-up spectroscopyof i-band drop-out objects. Cosmological GRBs are the ideal candidates for the universal study since several GRBs have been foundto locate at very high universe with redshifts larger than z ∼
6, e.g., GRB 080913 ( z = .
7, Greiner, J. et al. 2009), GRB 050904 (z = z = .
1, Salvaterra et al. 2009). In principle, superluminal motions should be detectableand similar for all the distant celestial bodies if only the true velocity of the outflow is close to the speed of light and the viewingangle θ is su ffi ciently small, due to the beaming e ff ect of special relativity (see also Zhang et al. 2007). In addition, this phenomenonwould be multiple-band observed if the receiver on ground is sensitive enough. However, studies on superluminal motions of thesesources proceed slowly till now owing to the lack of radio observations with high resolution.In statistics, roughly 15-20% of AGNs are radio-loud. The variability of radio-loud AGNs behaves much active over widewavelengths (Xie et al. 1994). Unlike the steep-spectrum for radio-quiet AGNs, the spectrum of radio-loud AGNs is generally flatbecause of the more beamed outflows at radio frequencies, but not in the optical and X-ray energy bands. This might imply thatradio-loud AGNs are jet-dominated at radio frequencies and disk-dominated in the X-ray and optical band. Even though, the radiomechanism of the radio-loud AGNs could be very di ff erent from that of LGRBs. The later with a single or double power-law formis thought to be basically produced from the processes of synchrotron radiation and inverse-Compton scattering. Currently, the realradiation mechanism of AGNs and GRBs are still not confirmed and needs to be further clarified. Acknowledgements.
This work is supported by the National Natural Science Foundation of China (grant number: U1431126, 11263002), Guizhou natural andscientific fundings (grant numbers: 20165660, 201519, OP201511 and 114A11KYSB20160008).
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Table 1.
Samples of Radio-Loud AGNs (B: Blazars; G: radio Galaxies; Q: Quassars)
IAU type z β app θ δ γ β +
285 B 0.102 2 36.41 1.56 2.38 0.907450219 +
428 B 0.444 14.98 5.8 1.99 12.34 0.996710235 +
164 B 0.94 7.1 2.57 16.32 9.74 0.994720454 +
844 B 0.3 1.6 8.49 4.46 2.63 0.924890716 +
714 B 0.3 2.3 22.33 2.63 2.51 0.917210735 +
178 B 0.424 5.84 6.39 3.17 7.27 0.990490851 +
202 B 0.306 3.2 3.16 10.33 5.71 0.984550954 +
658 B 0.367 5.7 8.61 6.62 5.84 0.985231101 +
384 B 0.031 1.9 20.2 2.78 2.22 0.89281308 +
326 B 0.996 20.7 4.74 8.45 29.65 0.999431749 +
701 B 0.77 6.12 17.76 1.32 15.18 0.997831803 +
784 B 0.684 3.9 3.56 10.55 6.04 0.98621823 +
568 B 0.664 2.56 17.35 3.29 2.79 0.933562007 +
776 B 0.342 2.3 6.73 5.89 3.48 0.957822200 +
420 B 0.069 3.7 10.24 5.32 4.04 0.968881323 +
321 G 0.729 1.7 59 0.8 1.73 0.816010108 +
388 G 0.669 2.14 40.66 1.18 2.95 0.940790710 +
439 G 0.518 1.25 68.73 0.64 2.32 0.902341845 +
797 G 0.0546 1.82 44.83 1.16 2.44 0.912162021 +
614 G 0.227 0.2 14.42 1.59 1.12 0.450340430 +
052 G 0.033 4 8.65 6.13 4.45 0.974420316 +
413 G 0.017 0.43 50.54 0.33 1.97 0.861580415 +
379 G 0.049 3.42 30.12 1.05 6.6 0.988451830 +
285 Q 0.594 2.55 38.36 0.97 4.35 0.973222223-052 Q 1.404 3.4 0.65 24.24 12.38 0.996731721 +
343 Q 0.206 2.3 46.64 0.23 13.6 0.997291253-055 Q 0.538 9.2 1.99 21.09 12.58 0.996841226 +
023 Q 0.158 8 8.48 6.66 8.21 0.992552230 +
114 Q 1.037 14.2 7.85 2.3 45.13 0.999751222 +
216 Q 0.435 1.4 21.49 2.48 1.84 0.839421828 +
487 Q 0.691 8.32 2.9 16.14 10.25 0.995231458 +
718 Q 0.905 6.53 4.79 10.68 7.38 0.990781633 +
382 Q 1.814 4.8 5.48 8.83 5.78 0.984921641 +
399 Q 0.595 9.5 8.32 6.37 10.35 0.995321618 +
177 Q 0.555 1.88 53.9 0.46 5.21 0.981411928 +
738 Q 0.302 7 10.29 5.4 7.33 0.990651611 +
343 Q 1.401 11.4 8.4 5.04 15.52 0.997921510-089* Q 0.361 3.77 2.31 13.18 7.17 0.990231606 + +
295 Q 0.729 26.1 4.03 7.85 47.38 0.999781150 +
812 Q 1.25 4.1 20.77 2.41 4.9 0.978951137 +
660 Q 0.652 1.3 65.29 0.71 1.45 0.724140420-00 Q 0.844 6.1 4.17 11.46 7.4 0.990830420-014* Q 0.915 4.8 3.45 11.72 6.89 0.989410458-020* Q 2.286 4.09 1.41 17.8 9.4 0.994330528 +
134 Q 2.07 5.15 2.59 14.22 8.08 0.992310538 +
498 Q 0.545 1.3 0.58 15.94 8.05 0.992250336-01 Q 0.852 8.9 2.32 19.01 11.61 0.996280333 +
321 Q 1.258 4.77 1.37 19.45 10.34 0.995310106 +
013 Q 2.107 8.2 1.54 23.34 13.34 0.997190153 +
744 Q 2.34 3.43 20.41 2.77 3.69 0.962580202 +
149 Q 0.833 0.4 1.34 5.93 3.06 0.945090212 +
735 Q 2.37 3.9 3.07 11.46 6.44 0.987870234 +
285 Q 1.213 9.29 2.19 19.99 12.18 0.996620552 +
398 Q 2.365 1.8 6.02 5.65 3.2 0.949920605-08* Q 0.872 4.4 12.75 4.53 4.51 0.975110923 +
392 Q 0.699 3.5 1.83 14.42 7.67 0.991460016 +
731 Q 1.781 8.3 4.03 12.95 9.17 0.994041039 +
811 Q 1.26 2.2 40.99 1.11 3.19 0.949591040 +
123 Q 1.029 3.1 24.76 2.2 3.51 0.958561055 + +
430 Q 0.67 3.86 0.17 51.64 25.97 0.999260850 +
581 Q 1.332 3.9 14.63 3.9 4.03 0.968720615 +
820 Q 0.71 2.2 24.3 2.43 2.42 0.910630711 +
356 Q 1.62 5.5 2.36 15.41 5.54 0.983570723 +
679 Q 0.846 4.8 23.3 0.5 24.35 0.999160827 +
243 Q 2.046 12.08 3.6 15.46 12.48 0.996780836 +
710 Q 2.17 10.4 5.51 10.41 10.45 0.995412251 ++